Complex numbers are numbers that consist of a real part and an imaginary part. The imaginary part is represented by the letter "i", which stands for the square root of -1. Complex numbers are written in the form a + bi, where a is the real part and bi is the imaginary part.
To add or subtract complex numbers, you simply add or subtract the real and imaginary parts separately. For example, to add (2 + 3i) and (4 + 5i), you would add 2 and 4 to get 6 as the real part, and add 3i and 5i to get 8i as the imaginary part. Therefore, the sum of these two complex numbers would be 6 + 8i.
The complex conjugate of a complex number is the number with the same real part, but with the sign of the imaginary part changed. For example, the complex conjugate of 3 + 4i is 3 - 4i. This is important when dividing complex numbers, as multiplying a complex number by its conjugate results in a real number.
To multiply complex numbers, you can use the FOIL method, just like you would with binomials. For example, to multiply (3 + 2i) and (5 + 4i), you would first multiply 3 and 5 to get 15 as the real part, then multiply 3 and 4i and 2i and 5 to get 12i and 10i respectively as the imaginary part. Therefore, the product of these two complex numbers would be 15 + 22i.
To divide complex numbers, you would multiply the numerator and denominator by the complex conjugate of the denominator. This results in a real number in the denominator, making it easier to divide.
To show unique solutions for complex numbers, you can use the quadratic formula. Simply treat the complex number as a quadratic equation, with the imaginary part as the coefficient of i. Then, solve for the value of i using the quadratic formula. The two solutions for i will give you the two unique solutions for the complex number.